We refer to Ref. [1]} for further details about the definitions of the basic chiral building tensors \(f_{\pm }^{\mu \nu }, u^\mu \)  and the assignments of the flavor contents of the resonance multiplets \(\hat{V}(\hat{A})_{\mu \nu }\) . The contributions from the minimal R\(\chi \) T to the \(\tau \rightarrow \pi \pi \gamma \nu _\tau \)  decay amplitudes have been evaluated in Ref. [2]}. Later on, a vector-meson-dominant (VMD) model was developed to include the refined effects from the odd-parity-intrinsic interacting vertex with \(\rho \omega \pi \) , together with other terms beyond the minimal R\(\chi \) T operators [3]}, [4]}. Compared to the results in Ref. [2]}, the deviations obtained in the latter two references are overwhelmingly caused by the intermediate \(\omega \)  meson. This conclusion is also confirmed by a recent study in Ref. [6]}, which extends the minimal R\(\chi \) T study by including a large amount of higher order odd- and even-intrinsic parity operators from Refs. [7]}, [8]}, respectively.
